Writing Truth Tables
1. Determining the rows in the truth table
The number of rows of a truth table is determined by the number of sentence letters in the formula or
argument. If n is the number of letters, then the number of rows is equal to 2n .
number of different letters in formula/argument
numb er of ro ws in truth table
1
2
2
4
3
8
4
16
5
32
6
64
etc.
To make sure all truth value combinations are covered in the truth table, assign the truth values to the
letters in alphabetical order (this is not logically necessary but necessary for grading!!):
If m is the number of rows necessary,
1. assign to the first letter (in alphabetical order), T m/2 times and F m/2 times
2. to the next letter, T m/4 times and F m/4 times and again T m/4 times and F m/4 times
3. to the third letter, T m/8 times and F m/8 times, repeat three more times,
4. continue until the last letter gets assigned T, F, T, F, alternating for every row.
2. Memorizing truth tables for the basic connectives
In general, students find it easier to memorize English rules for writing the truth tables than to memorize
the truth tables themselves.
Connective/Meaning
Rule
~ not
flip the truth value to the right (i.e. T to F, F to T)
& and
write T if both sides are T,
write F otherwise (i.e. if at least one side is F)
v or
write F if both sides are F,
write T otherwise (i.e. if at least one side is T)
-> if..then
write F if (the left side is T and the right side is F),
write T otherwise (i.e. if either the left side is F or the right side is T or
both)
<-> if and only if
write T if the two sides have the same truth value,
write F otherwise (i.e. if the two sides have different truth values)
3. Filling in the truth table for a proposition
Fill in the truth value for the partial formulas beginning with the innermost parentheses or the negation.
Note that the parentheses around a negation are usually not written as it is understood that negation
applies first, just like a negative sign in math.
You can either use a new column for each partial formula (clearer; this is what Dr. Burch has been doing
in class) or write the truth values under the connective you are working on (less space; this is what the
book does).
Example: ((Q & P) v ~(Q ->P)) <-> P
1. Determine the number of rows: there are two letters, so there are 22 = 4 rows.
2. Assign the truth values to the letters in alphabetical order from top to bottom (P: TTFF, Q: TFTF)
3. Determine the order in which the connectives need to be worked on:
a. & because it is in parentheses
b. -> because it is in parentheses
c. ~ because the truth values of (P ->Q) is known and negation comes next
d. v because it is in parentheses
e. <-> last connective
4. Fill in the values in the order determined according to the rules for the connectives.
Partial Solution (the connective worked on is bolded):
a. in an extended truth table in which the partial formulas are written in separate columns
P
Q
Q&P
Q ->P
~(Q ->P)
T
T
T
T
F
T
F
F
T
F
F
T
F
F
T
F
F
2 and 1
2 and 1
4
use columns:
(Q & P) v ~(Q >P)
((Q & P) v ~(Q ->P))
<-> P
3 and 5
6 and 1
b. in one table
formula
filling order
(( Q
&
P)
v
~
(Q
->
P ))
<->
P
1
2
1
5
4
1
3
1
6
1
T
T
T
F
T
T
T
T
F
F
T
F
F
T
T
T
T
F
F
T
T
F
F
F
F
F
F
F
F
The last column filled in (here column 6) determines the truth values of the proposition expressed by the
formula. This column allows classifcation of the proposition. If this column contains only T, the
proposition is tautologous. If this column contains only F, the proposition is self-contradictory.
Otherwise (at least one T, at least one F), the proposition is contingent.
4. Filling in truth tables for several propositions
Determine the rows of the truth table as before. Then determine the truth values for all propositions as
before. Work on each proposition independently but in the same table. They have the assignment of the
truth values to the letters in common. If they also have partial formulas in common you may use those
columns for both/all propositions.
Each proposition’s truth values are determined by the column of the last connective filled in (the main
connective or main operator). The relation between pairs of propositions can be classified according to
these columns (assuming you have constructed the truth table correctly, of course!).
Truth values of proposition (column of main
operator)
Relationship
Example
same truth value in every row
logically equivalent
Pv~Q and ~P ->~Q
opposite truth value in every row (if one is T, the
other one is F and vice versa)
contradictory
~(Q ->P) and ~Q vP
in at least one row both have T
consistent
P ->Q and Q ->P
no row in which both have T
inconsistent, contrary
Q & ~P and P & ~Q
5. Filling in truth tables for an argument
Proceed as in section 4 for each proposition in the argument, i.e. for all premises and for the conclusion.
6. Evaluating an argument as valid or invalid.
Construct a truth table as in section 5. Then try to find a row in which all premises have the value T
and the conclusion has the value F. If there is at least one, the argument is invalid; if there is no such
row, the argument is valid (assuming you have constructed the truth table correctly, of course!).